For instance, let's say an EMT goes to a car accident and in the process of rendering aid, poorly splints an injured limb causing nerve damage. So the patient sues. Was the EMT within scope of employment, yes. Should he be granted coverage by the department in the suit? I would say yes. Should his own liability policy have to be used? No.
Another EMT from the same department goes to the scene of an accident and the patient is suffering from a tension pneumothorax. The fire department doesn't allow for needle decompression, but the EMT saw it done once during a combat tour in Afghanistan and gives it a shot. The patient succumbs to his injuries and the family sues. Was the EMT within scope? No, he performed a procedure not authorized by his department. Who pays for the legal defense? He does from his own liability policy.
How are MDs “practicing professionals” while RNs are “hired by an organization that has granted them authority to act in their name?” It depends on who you work for.
Personally one of the things I would like to see in a second bill of rights is a requirement that all government employees be subject to the same civil liability as the rest of us. The American revolution was about getting rid of a sovereign and replacing him with rule by the people with protections for individual rights. Then of course the court draw from British common law and stick us with sovereign immunity.
The big one you don't mention is that claims against public sector workers have statutory limits.